Part 1: Particles and waves

Q1. What is a particle?

A particle is a discrete unit of matter having the attributes of
mass, momentum (and thus kinetic energy) and optionally of
electric charge.

Q2. What is a wave?

A wave is a periodic variation of some quantity as a function of location
or time. For example, the wave motion of a vibrating guitar string is
defined by the displacement of the string from its center as a function of
distance along the string. A sound wave consists of variations in the pressure
with location.

A wave is characterized by its wavelength l
(lambda) and frequency n (nu), which are
connected by the relation

in which u is the velocity of propagation of the disturbance in
the medium.

Problem example: The velocity of sound in the air is 330 m
s–1. What is the wavelength of A440 on the piano keyboard?

Solution:

Two other attributes of waves are the amplitude (the height of the
wave crests with respect to the base line) and the phase, which
measures the position of a crest with respect to some fixed point. The square of
the amplitude gives the intensity of the wave: the energy transmitted
per unit time).

A unique property of waves is their ability to combine constructively or
destructively, depending on the relative phases of the combining waves.

Q3. What is light?

Phrasing the question in this way reflects the deterministic mode of Western
thought which assumes that something cannot "be" two quite different things at
the same time. The short response to this question is that all we know about
light (or anything else, for that matter) are the results of experiments, and
that some kinds of experiments show that light behaves like particles,
and that other experiments reveal light to have the properties of waves.
For the moment, it is better to amend this question to

Q4. What is the wave theory of light?

In the early 19th century, the English scientist Thomas Young carried out the
famous two-slit experiment which demonstrated that a beam of light, when split
into two beams and then recombined, will show interference effects that can
only be explained by assuming that light is a wavelike disturbance. By 1820,
Augustin Fresnel had put this theory on a sound mathematical basis, but the
exact nature of the waves remained unclear until the 1860's when James Clerk
Maxwell developed his electromagnetic theory.

From the laws of electromagnetic induction that were discovered in the period
1820-1830 by Hans Christian Oersted and Michael Faraday, it was known that a
moving electric charge gives rise to a magnetic field, and that a changing magnetic
field can induce electric charges to move. Maxwell showed theoretically that
when an electric charge is accelerated (by being made to oscillate within a
piece of wire, for example), electrical energy will be lost, and an equivalent
amount of energy is radiated into space, spreading out as a series of waves
extending in all directions.

What is "waving" in electromagnetic radiation? According to
Maxwell, it is the strengths of the electric and magnetic fields as they
travel through space. The two fields are oriented at right angles to each
other and to the direction of travel.

As the electric field changes, it induces a magnetic field,
which then induces a new electric field, etc., allowing the wave to
propagate itself through space

These waves consist of periodic variations in the electrostatic and electromagnetic
field strengths. These variations occur at right angles to each other. Each
electrostatic component of the wave induces a magnetic component, which then
creates a new electrostatic component, so that the wave, once formed, continues
to propagate through space, essentially feeding on itself. In one of the most
brilliant mathematical developments in the history of science, Maxwell expounded
a detailed theory, and even showed that these waves should travel at about 3E8 m s–1,
a value which experimental observations had shown corresponded to the speed
of light. In 1887, the German physicist Heinrich Hertz demonstrated that an
oscillating electric charge (in what was in essence the world's first radio
transmitting antenna) actually does produce electromagnetic radiation just as
Maxwell had predicted, and that these waves behave exactly like light.

It is now understood that light is electromagnetic radiation that
falls within a range of wavelengths that can be perceived by the eye. The entire
electromagnetic spectrum runs from radio waves at the long-wavelength end,
through heat, light, X-rays, and to gamma radiation.

Part 2: Quantum theory of light

Q5. How did the quantum theory of light come
about?

It did not arise from any attempt to explain the behavior of light itself; by
1890 it was generally accepted that the electromagnetic theory could explain all
of the properties of light that were then known.

Certain aspects of the interaction between light and matter that were observed
during the next decade proved rather troublesome, however. The relation between
the temperature of an object and the peak wavelength emitted by it was established
empirically by Wilhelm Wien in 1893. This put on a quantitative basis what everyone
knows: the hotter the object, the "bluer" the light it emits.

Q6. What is black body radiation?

All objects above the temperature of absolute zero emit electromagnetic radiation
consisting of a broad range of wavelengths described by a distribution curve
whose peak wavelength l at absolute temperature T
for a "perfect radiator" known as a black body is given by Wien's law

At ordinary temperatures this radiation
is entirely in the infrared region of the spectrum, but as the temperature rises
above about 1000K, more energy is emitted in the visible wavelength region and
the object begins to glow, first with red light, and then shifting toward the
blue as the temperature is increased.

This type of radiation has two important characteristics. First,
the spectrum is a continuous one, meaning that all wavelengths are emitted,
although with intensities that vary smoothly with wavelength. The other curious
property of black body radiation is that it is independent of the composition of
the object; all that is important is the temperature.

Q7. How did black body radiation lead to quantum
physics?

Black body radiation, like all electromagnetic radiation, must originate from
oscillations of electric charges which in this case were assumed to be the
electrons within the atoms of an object acting somewhat as miniature Hertzian
oscillators. It was presumed that since all wavelengths seemed to be present in
the continuous spectrum of a glowing body, these tiny oscillators could send or
receive any portion of their total energy. However, all attempts to predict the
actual shape of the emission spectrum of a glowing object on the basis of
classical physical theory proved futile.

In 1900, the great German physicist Max Planck (who earlier
in the same year had worked out an empirical formula giving the detailed shape
of the black body emission spectrum) showed that the shape of the observed spectrum
could be exactly predicted if the energies emitted or absorbed by each oscillator
were restricted to integral values of hn, where n
("nu") is the frequency and h is a constant 6.626E–34 J s which we
now know as Planck's Constant. The allowable energies of each oscillator are
quantized, but the emission spectrum of the body remains continuous because
of differences in frequency among the uncountable numbers of oscillators it
contains.This modification of classical theory, the first use of the quantum
concept, was as unprecedented as it was simple, and it set the stage for the
development of modern quantum physics.

Q8. What is the photoelectric effect?

Shortly after J.J. Thompson's experiments led to the identification of
the elementary charged particles we now know as electrons, it was discovered
that the illumination of a metallic surface by light can cause electrons
to be emitted from the surface. This phenomenon, the photoelectric effect,
is studied by illuminating one of two metal plates in an evacuated tube.
The kinetic energy of the photoelectrons causes them to move to the opposite
electrode, thus completing the circuit and producing a measurable current.
However, if an opposing potential (the retarding potential) is imposed
between the two electrons, the kinetic energy can be reduced to zero so
that the electron current is stopped. By observing the value of the retarding
potential Vr, the kinetic energy of the photoelectrons can
be calculated from the electron charge e, its mass m and the frequency
of the incident light:

These two diagrams are taken from a Web page
by Joseph Alward of the University of the Pacific.

The plot at the right shows how the kinetic energy of the
photoelectrons falls to zero at the critical wavelength corresponding to
frequency f0.

Q9. What peculiarity of the photoelectric effect led
to the photon?

Although the number of electrons ejected from the metal surface per second
depends on the intensity of the light, as expected, the kinetic energies of
these electrons (as determined by measuring the retarding potential needed to
stop them) does not, and this was definitely not expected. Just as a more
intense physical disturbance will produce higher energy waves on the surface of
the ocean, it was supposed that a more intense light beam would confer greater
energy on the photoelectrons. But what was found, to everyone's surprise, is
that the photoelectron energy is controlled by the wavelength of the light, and
that there is a critical wavelength below which no photoelectrons are emitted at
all.

Albert Einstein quickly saw that if the kinetic energy of
the photoelectrons depends on the wavelength of the light, then so must its
energy. Further, if Planck was correct in supposing that energy must be exchanged
in packets restricted to certain values, then light must similarly be organized
into energy packets. But a light ray consists of electric and magnetic fields
that spread out in a uniform, continuous manner; how can a continuously-varying
wave front exchange energy in discrete amounts? Einstein's answer was that the
energy contained in each packet of the light must be concentrated into a tiny
region of the wave front. This is tantamount to saying that light has the nature
of a quantized particle whose energy is given by the product of Planck's constant
and the frequency:

Einstein's publication of this explanation in 1905 led to the rapid
acceptance of Planck's idea of energy quantization, which had not previously
attracted much support from the physics community of the time. It is interesting
to note, however, that this did not make Planck happy at all. Planck, ever the
conservative, had been reluctant to accept that his own quantized-energy
hypothesis was much more than an artifice to explain black-body radiation; to
extend it to light seemed an absurdity that would negate the well-established
electromagnetic theory and would set science back to the time before Maxwell.

Q10. Where does relativity come in?

Einstein's special theory of relativity arose from his attempt to understand
why the laws of physics that describe the current induced in a fixed conductor
when a magnet moves past it are not formulated in the same way as the ones that
describe the magnetic field produced by a moving conductor. The details of this
development are not relevant to our immediate purpose, but some of the conclusions
that this line of thinking led to very definitely are. Einstein showed that
the velocity of light, unlike that of a material body, has the same value no
matter what velocity the observer has. Further, the mass of any material object,
which had previously been regarded as an absolute, is itself a function of the
velocity of the body relative to that of the observer (hence "relativity"),
the relation being given by

in which mo is the rest mass of the particle, v is its velocity
with respect to the observer, and c is the velocity of light.

According to this formula, the mass of an object increases without limit as
the velocity approaches that of light. Where does the increased mass come from?
Einstein's answer was that the increased mass is that of the kinetic energy of
the object; that is, energy itself has mass, so that mass and energy are
equivalent according to the famous formula

The only particle that can move at the velocity of light is the photon
itself, due to its zero rest mass.

Q11. Can the mass-less photon have momentum?

Although the photon has no rest mass, its energy, given by , confers upon it
an effective mass of

and a momentum of

Q12. If waves can be particles, can particles be waves?

In 1924, the French physicist Louis deBroglie proposed (in
his doctoral thesis) that just as light possesses particle-like properties,
so should particles of matter exhibit a wave-like character. Within two years
this hypothesis had been confirmed experimentally by observing the diffraction
(a wave interference effect) produced by a beam of electrons as they were scattered
by the row of atoms at the surface of a metal.

deBroglie showed that the wavelength of a particle is inversely proportional
to its momentum:

Notice that the wavelength of a stationary particle is infinitely large,
while that of a particle of large mass approaches zero. For most practical
purposes, the only particle of interest to chemistry that is sufficiently small
to exhibit wavelike behavior is the electron (mass 9.11E­31 kg).

Q13. Exactly what is it that is "waving"?

We pointed out earlier that a wave is a change that varies with location in a
periodic, repeating way. What kind of a change do the crests and hollows of a
"matter wave" trace out? The answer is that the wave represents the value of a
quantity whose square is a measure of the probability of finding the particle in
that particular location. In other words, what is "waving" is the value of a
mathematical probability function.

Q14. What is the uncertainty principle?

In 1927, Werner Heisenberg proposed that certain pairs of
properties of a particle cannot simultaneously have exact values. In particular,
the position and the momentum of a particle have associated with them uncertainties
x and p given by

As with the de Broglie particle wavelength, this has practical consequences
only for electrons and other particles of very small mass. It is very important
to understand that these "uncertainties" are not merely limitations related
to experimental error or observational technique, but instead they express an
underlying fact that Nature does not allow a particle to possess definite values
of position and momentum at the same time. This principle (which would be better
described by the term "indeterminacy" than "uncertainty") has been thoroughly
verified and has far-reaching practical consequences which extend to chemical
bonding and molecular structure.

Q15. Is the uncertainty principle consistent with particle
waves?

Yes; either one really implies the other. Consider the following two limiting
cases: · A particle whose velocity is known to within a very small uncertainty
will have a sharply-defined energy (because its kinetic energy is known) which
can be represented by a probability wave having a single, sharply-defined
frequency. A "monochromatic" wave of this kind must extend infinitely in
space:

But if the peaks of the wave represent locations at which the particle is
most likely to manifest itself, we are forced to the conclusion that it can "be"
virtually anywhere, since the number of such peaks is infinite! Now think of the
opposite extreme: a particle whose location is closely known. Such a particle
would be described by a short wavetrain having only a single peak, the smaller
the uncertainty in position, the more narrow the peak.

To help you see how waveforms of different wavelength combine, two such
combinations are shown below:

It is apparent that as more waves of different frequency are mixed, the
regions in which they add constructively diminish in extent. The extreme case
would be a wavetrain in which destructive interference occurs at all locations
except one, resulting in a single pulse:

Is such a wave possible, and if so, what is its wavelength? Such a wave is
possible, but only as the sum (interference) of other waves whose wavelengths
are all slightly different. Each component wave possesses its own energy (momentum),
and adds that value to the range of momenta carried by the particle, thus increasing
the uncertainty p. In the extreme case of a quantum particle whose location
is known exactly, the probability wavelet would have zero width which could
be achieved only by combining waves of all wavelengths-- an infinite number
of wavelengths, and thus an infinite range of momentum dp and thus kinetic energy.

Q16. Are they particles or are they waves?

Suppose we direct a beam of photons (or electrons; the experiment works with
both) toward a piece of metal having a narrow opening. On the other side there
are two more openings, or slits. Finally the particles impinge on a photographic
plate or some other recording device. Taking into account their wavelike
character, we would expect the probability waves to produce an interference
pattern of the kind that is well known for sound and light waves, and this is
exactly what is observed; the plate records a series of alternating dark and
light bands, thus demonstrating beyond doubt that electrons and light have the
character of waves.

Now let us reduce the intensity of the light so that only one photon at a
time passes through the apparatus (it is experimentally possible to count single
photons, so this is a practical experiment). Each photon passes through the
first slit, and then through one or the other of the second set of slits,
eventually striking the photographic film where it creates a tiny dot. If we
develop the film after a sufficient number of photons have passed through, we
find the very same interference pattern we obtained previously.

There is something strange here. Each photon, acting as a particle, must pass
through one or the other of the pair of slits, so we would expect to get only
two groups of spots on the film, each opposite one of the two slits. Instead, it
appears that the each particle, on passing through one slit, "knows" about the
other, and adjusts its final trajectory so as to build up a wavelike
interference pattern.

It gets even stranger: suppose that we set up a detector to determine which
slit a photon is heading for, and then block off the other slit with a shutter.
We find that the photon sails straight through the open slit and onto the film
without trying to create any kind of an interference pattern. Apparently, any
attempt to observe the photon as a discrete particle causes it to behave like
one.

The only conclusion possible is that quantum particles have no well defined
paths; each photon (or electron) seems to have an infinity of paths which thread
their way through space, seeking out and collecting information about all possible
routes, and then adjusting its behavior so that its final trajectory, when combined
with that of others, produces the same overall effect that we would see from
a train of waves of wavelength = h/mv.

Part 3: Electrons in atoms

Q17. What are line spectra?

We have already seen that a glowing body (or actually, any body whose
temperature is above 0K) emits and absorbs radiation of all wavelength in a
continuous spectrum. In striking contrast is the spectrum of light
produced when certain substances are volatilized in a flame, or when an electric
discharge is passed through a tube containing gaseous atoms of an element. The
light emitted by such sources consists entirely of discrete wavelengths. This
kind of emission is known as a discrete spectrum or line spectrum
(the "lines" that appear on photographic images of the spectrum are really
images of the slit through which the light passes before being dispersed by the
prism in the spectrograph).

Every element has its own line spectrum which serves as a sensitive and
useful tool for detecting the presence and relative abundance of the element,
not only in terrestrial samples but also in stars. (As a matter of fact, the
element helium was discovered in the sun, through its line spectrum, before it
had been found on Earth.) In some elements, most of the energy in the visible
part of the emission spectrum is concentrated into just a few lines, giving
their light characteristic colors: yellow-orange for sodium, blue-green for
mercury (these are commonly seen in street lights) and orange for neon.

Line spectra were well known early in the 19th century, and were widely used
for the analysis of ores and metals. The German spectroscopist R.W. Bunsen,
now famous for his gas burner, was then best known for discovering two new elements,
rubidium and cesium, from the line spectrum he obtained from samples of mineral
spring waters.

Q18. How are line spectra organized?

Until 1885, line spectra were little more than "fingerprints" of the elements;
extremely useful in themselves, but incapable of revealing any more than the
identify of the individual atoms from which they arise. In that year a Swiss
school teacher named Johann Balmer published a formula that related the wavelengths
of the four known lines in the emission spectrum of hydrogen in a simple way.
Balmer's formula was not based on theory; it was probably a case of cut-and-try,
but it worked: he was able to predict the wavelength of a fifth, yet-to-be discovered
emission line of hydrogen, and as spectroscopic and astronomical techniques
improved (the only way of observing highly excited hydrogen atoms at the time
was to observe the solar spectrum during an eclipse), a total of 35 lines were
discovered, all having wavelengths given by the formula which we write in the
modern manner as

in which m = 2 and R is a constant (the Rydberg constant, after the Swedish
spectroscopist) whose value is 1.09678E7 m–1. The variable n is
an integer whose values 1, 2, etc. give the wavelengths of the different
lines.

It was soon discovered that by replacing m with integers other than 2, other
series of hydrogen lines could be accounted for. These series, which span the
wavelength region from the ultraviolet through infrared, are named after their
discoverers.

name of series

when discovered

value of m

Lyman

1906-14

1

Balmer

1885

2

Paschen

1908

3

Brackett

1922

4

Pfund

1924

5

Attempts to adapt Balmer's formula to describe the spectra of atoms other
than hydrogen generally failed, although certain lines of some of the spectra
seemed to fit this same scheme, with the same value of R.

Q19. How large can n be?

There is no limit; values in the hundreds have been observed, although doing
so is very difficult because of the increasingly close spacing of successive
levels as n becomes large. Atoms excited to very high values of n are said to
be in Rydberg states.

Q20. Why do line spectra become continuous at short
wavelengths?

As n becomes larger, the spacing between neighboring levels diminishes and
the discrete lines merge into a continuum. This can mean only one thing:
the energy levels converge as n approaches infinity. This convergence limit
corresponds to the energy required to completely remove the electron from the
atom; it is the ionization energy.

At energies in excess of this, the electron is no longer bound to the rest of
the atom, which is now of course a positive ion. But an unbound system is not
quantized; the kinetic energy of the ion and electron can now have any value in
excess of the ionization energy. When such an ion and electron pair recombine to
form a new atom, the light emitted will have a wavelength that falls in the
continuum region of the spectrum. Spectroscopic observation of the convergence
limit is an important method of measuring the ionization energies of atoms.

Q21. What were the problems with the planetary model of the
atom?

Rutherford's demonstration that the mass and the positive charge of the atom
is mostly concentrated in a very tiny region called the nucleus forced the question
of just how the electrons are disposed outside the nucleus. By analogy with
the solar system, a planetary model was suggested: if the electrons were orbiting
the nucleus, there would be a centrifugal force that could oppose the electrostatic
attraction and thus keep the electrons from falling into the nucleus. This of
course is similar to the way in which the centrifugal force produced by an orbiting
planet exactly balances the force due to its gravitational attraction to the
sun.

This model suffers from one fatal weakness: electrons, unlike planets, are
electrically charged. An electric charge revolving in an orbit is continually
undergoing a change of direction, that is, acceleration. It has been well known
since the time of Hertz that an accelerating electric charge radiates energy. We
would therefore expect all atoms to act as miniature radio stations. Even worse,
conservation of energy requires that any energy that is radiated must be at the
expense of the kinetic energy of the orbital motion of the electron. Thus the
electron would slow down, reducing the centrifugal force and allowing the
electron to spiral closer and closer to the nucleus, eventually falling into it.
In short, no atom that operates according to the planetary model would last long
enough for us to talk about it.

As if this were not enough, the planetary model was totally unable to explain
any of the observed properties of atoms, including their line spectra.

Q22. How did Bohr's theory save the planetary model... for a
while?

Niels Bohr was born in the same year (1885) that Balmer published his
formula for the line spectrum of hydrogen. Beginning in 1913, the brilliant
Danish physicist published a series of papers that would ultimately derive Balmer's
formula from first principles.

Bohr's first task was to explain why the orbiting electron does not radiate
energy as it moves around the nucleus. This energy loss, if it were to occur at
all, would do so gradually and smoothly. But Planck had shown that black body
radiation could only be explained if energy changes were limited to jumps
instead of gradual changes. If this were a universal characteristic of energy-
that is, if all energy changes were quantized, then very small changes in energy
would be impossible, so that the electron would in effect be "locked in" to its
orbit.

From this, Bohr went on to propose that there are certain stable orbits in
which the electron can exist without radiating and thus without falling into a
"death spiral". This supposition was a daring one at the time because it was
inconsistent with classical physics, and the theory which would eventually lend
it support would not come along until the work of de Broglie and Heisenberg
more than ten years later.

Since Planck's quanta came in multiples of h, Bohr restricted his allowed
orbits to those in which the product of the radius r and the momentum of the
electron mv (which has the same units as h, J-s) are integral multiples of
h:

2rmv = nh (n =
1,2,3, . .)

Each orbit corresponds to a different energy, with the electron normally
occupying the one having the lowest energy, which would be the innermost orbit
of the hydrogen atom.

Taking the lead from Einstein's explanation of the photoelectric effect, Bohr
assumed that each spectral line emitted by an atom that has been excited by
absorption of energy from an electrical discharge or a flame represents a change
in energy given by E = hn = hc/, the energy lost when the electron falls from a higher
orbit (value of n) into a lower one.

Finally, as a crowning triumph, Bohr derived an expression giving the radius
of the nth orbit for the electron in hydrogen as

Substitution of the observed values of the electron mass and electron charge
into this equation yielded a value of 0.529E­10 m for the radius of the
first orbit, a value that corresponds to the radius of the hydrogen atom
obtained experimentally from the kinetic theory of gases. Bohr was also able to
derive a formula giving the value of the Rydberg constant, and thus in effect
predict the entire emission spectrum of the hydrogen atom.

Q23. What were the main problems with Bohr's theory?

There were two kinds of difficulties. First, there was the practical
limitation that it only works for atoms that have one electron-- that is, for H,
He+, Li2+, etc. The second problem was that Bohr was
unable to provide any theoretical justification for his assumption that
electrons in orbits described by the preceding equation would not lose energy by
radiation. This reflects the fundamental underlying difficulty: because
de Broglie's picture of matter waves would not come until a decade later,
Bohr had to regard the electron as a classical particle traversing a definite
orbital path.

Q24. How did the wave picture of the electron save Bohr's
theory?

Once it became apparent that the electron must have a wavelike character,
things began to fall into place. The possible states of an electron confined to
a fixed space are in many ways analogous to the allowed states of a vibrating
guitar string. These states are described as standing waves that must possess
integral numbers of nodes. The states of vibration of the string are described
by a series of integral numbers n = 1,2,... which we call the
fundamental, first overtone, second overtone, etc. The energy of vibration is
proportional to n2. Each mode of vibration contains one more complete
wave than the one below it.

In exactly the same way, the mathematical function that defines the
probability of finding the electron at any given location within a confined
space possesses n peaks and corresponds to states in which the energy is
proportional to n2.

The electron in a hydrogen atom is bound to the nucleus by its spherically
symmetrical electrostatic charge, and should therefore exhibit a similar kind of
wave behavior. This is most easily visualized in a two-dimensional cross section
that corresponds to the conventional electron orbit. But if the particle picture
is replaced by de Broglie's probability wave, this wave must follow a
circular path, and- most important of all- its wavelength (and consequently its
energy) is restricted to integral multiples n = 1,2,.. of the
circumference

for otherwise the wave would collapse owing to self-interference. That is,
the energy of the electron must be quantized; what Bohr had taken as a daring
but arbitrary assumption was now seen as a fundamental requirement. Indeed the
above equation can be derived very simply by combining Bohr's quantum condition
2prmv = nh with the expression mv = h/l for the deBroglie wavelength of a particle.

Viewing the electron as a standing-wave pattern also explains its failure to
lose energy by radiating. Classical theory predicts that an accelerating
electric charge will act as a radio transmitter; an electron traveling around a
circular wire would certainly act in this way, and so would one rotating in an
orbit around the nucleus. In a standing wave, however, the charge is distributed
over space in a regular and unchanging way; there is no motion of the charge
itself, and thus no radiation.

Q25. What is an orbital?

Because the classical view of an electron as a localizable particle is now
seen to be untenable, so is the concept of a definite trajectory, or orbit.
Instead, we now use the word orbital to describe the state of existence of an
electron. An orbital is really no more than a mathematical function describing
the standing wave that gives the probability of the electron manifesting itself
at any given location in space. More commonly (and loosely) we use the word to
describe the region of space occupied by an electron. Each kind of orbital is
characterized by a set of quantum numbers n, l, and m These relate,
respectively, to the average distance of the electron from the nucleus, to the
shape of the orbital, and to its orientation in space.

Q26. If the electron cannot be localized, can it be moving?

In its lowest state in the hydrogen atom (in which l=0) the electron has zero
angular momentum, so electrons in s orbitals are not in motion. In orbitals for
which l>0 the electron does have an effective angular momentum, and since the
electron also has a definite rest mass me = 9.11E­31 kg, it
must possess an effective velocity. Its value can be estimated from the
Uncertainty Principle; if the volume in which the electron is confined is about
10–10 m, then the uncertainty in its momentum is at least
h/(1010) = 6.6E–24 kg m s–1,
which implies a velocity of around 107 m s–1, or
almost one-tenth the velocity of light.

The stronger the electrostatic force of attraction by the nucleus, the faster
the effective electron velocity. In fact, the innermost electrons of the heavier
elements have effective velocities so high that relativistic effects set in;
that is, the effective mass of the electron significantly exceeds its rest mass.
This has direct chemical effects; it is the cause, for example, of the low
melting point of metallic mercury and of the color of gold.

Q27. Why does the electron not fall into the nucleus?

The negatively-charged electron is attracted to the positive charge of the
nucleus. What prevents it from falling in? This question can be answered in
various ways at various levels. All start with the statement that the electron,
being a quantum particle, has a dual character and cannot be treated solely by
the laws of Newtonian mechanics.

We saw above that in its wavelike guise, the electron exists as a standing
wave which must circle the nucleus at a sufficient distance to allow at least
one wavelength to fit on its circumference. This means that the smaller the
radius of the circle, the shorter must be the wavelength of the electron, and
thus the higher the energy. Thus it ends up "costing" the electron energy if it
gets too close to the nucleus. The normal orbital radius represents the balance
between the electrostatic force trying to pull the electron in, and what we
might call the "confinement energy" that opposes the electrostatic energy. This
confinement energy can be related to both the particle and wave character of the
electron.

If the electron as a particle were to approach the nucleus, the uncertainty
in its position would become so small (owing to the very small volume of space
close to the nucleus) that the momentum, and therefore the energy, would have to
become very large. The electron would, in effect, be "kicked out" of the nuclear
region by the confinement energy.

The standing-wave patterns of an electron in a box can be calculated quite
easily. For a spherical enclosure of diameter d, the energy is given by

in which n = 1,2,3. etc.

Q28. What is electron spin?

Each electron in an atom has associated with it a magnetic field whose
direction is quantized; there are only two possible values that point in
opposite directions. We usually refer to these as "up" and "down", but the
actual directions are parallel and antiparallel to the local magnetic field
associated with the orbital motion of the electron.

The term spin implies that this magnetic moment is produced by the electron
charge as the electron rotates about its own axis. Although this conveys a vivid
mental picture of the source of the magnetism, the electron is not an extended
body and its rotation is meaningless. Electron spin has no classical counterpart;
the magnetic moment is a consequence of relativistic shifts in local space and
time due to the high effective velocity of the electron in the atom. This effect
was predicted theoretically by P.A.M. Dirac in 1928.